Decision-making under conditions of risk and uncertainty

Decision-making under conditions of risk and uncertainty Solutions to Chapter 12 questions (a) Profit and Loss Statement for Period Ending 31 May 2000 Revenue (14 400 000 journeys): 0 3 miles (7 200 000 0.20) 1 440 000) 4 5 miles (4 320 000 0.30) 1 296 000) Over 5 miles (2 880 000 0.50) 1 440 000) Juvenile fares (4 800 000 0.15) 720 000) Senior citizen fares (4 800 000 0.10) 480 000) 5 376 000) Advertising revenue 250 000) 5 626 000) Less: Variable costs (20 routes 4 buses 150 miles 330 days 0.75) (2 970 000) Less: Fixed costs (1 750 000) Net profit 906 000) Question 12.17 (b) Assuming the same passenger mix as 2000 the weighted average fare per passenger for year ending 31 May 2001 is ( 5 376 000 1.05)/24 000 000 0.2352. The break-even point is where: Total revenue from fares Advertising revenue Total cost Let x number of passenger journeys Break-even point: 0.2352x 250 000 (2 970 000 1 750 000) 1.1 0.2352x 4 942 000 x 21 011 905 Maximum capacity utilization 40 000 000 passenger journeys (24 000 000/0.6) Break-even capacity utilization 21 011 905/40 000 000 52.5% (c) (i) Expected value and probability estimates for 2001 Capacity Revenue Inflation Costs Combined Net Expected Utilization Fares Adverts probability profit value % (Probability) ( 000) ( 000) (%) (Probability) ( 000) ( 000) ( 000) 70 0.1 6585.6 a 250 8 0.3 5097.6 b 0.03 1738.0 52.14 6585.6 250 10 0.6 5192.0 b 0.06 1643.6 98.62 6585.6 250 12 0.1 5286.4 b 0.01 1549.2 15.49 60 0.5 5644.8 a 250 8 0.3 5097.6 0.15 797.2 119.58 5644.8 250 10 0.6 5192.0 0.30 702.8 210.84 5644.8 250 12 0.1 5286.4 0.05 608.4 30.42 50 0.4 4704.0 a 250 8 0.3 5097.6 0.12 143.6 17.23 4704.0 250 10 0.6 5192.0 0.24 238.0 57.12 4704.0 250 12 0.1 5286.4 0.04 332.4 13.30 1.00 439.44 DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY 95

Notes a Fare revenues at 60% capacity for 2000 were 5 376 000. Assuming 5% inflation fare revenues for 2001 at 60% capacity will be 5 644 800 ( 5 376 000 1.05). At 70% and 50% capacity utilization fare revenues will be as follows: 70% 70/60 5 644 800 6 585 600 50% 50/60 5 644 800 4 704 000 b Variable costs vary with bus miles which are assumed to remain unchanged. Predicted costs at the different inflation levels are as follows: 8% ( 2 970 000 1 750 000)1.08 5 097 600 10% ( 2 970 000 1 750 000)1.10 5 192 000 12% ( 2 970 000 1 750 000)1.12 5 286 400 (c) (ii) The answer to this question requires the preparation of a cumulative probability distribution that measures the cumulative probability of profits/ (losses) being greater than specified levels. Cumulative probability distribution Losses greater than 300 000 0.04 probability Probability of a loss occurring 0.40 Profits greater than 600 000 0.60 Profits greater than 700 000 0.55 Profits greater than 800 000 0.10 Profits greater than 1 500 000 0.10 (d) The following factors have not been incorporated into the analysis: (i) Change in the passenger mix. (ii) Changes in the number of routes and the number of days operation per year. (iii) Changes in fare structure such as off-peak travel or further concessions for juveniles and senior citizens. (iv) Changes in cost levels due to factors other than inflation (e.g. more efficient operating methods). Question 12.18 (a) For each selling price there are three possible outcomes for sales demand, unit variable cost and fixed costs. Consequently, there are 27 possible outcomes. In order to present probability distributions for the two possible selling prices, it would be necessary to compute profits for 54 outcomes. Clearly, there would be insufficient time to perform these calculations within the examination time that can be allocated to this question. It is therefore assumed that the examiner requires the calculations to be based on an expected value approach. The expected value calculations are as follows: (i) Variable cost (ii) Fixed costs ( 10 10%) 10/20 5.50 82 000 0.3 24 600 ( 10 6/20 3.00 85 000 0.5 42 500 ( 10 5%) 4/20 1.90 90 000 0.2 18 000 10.40 85 100 (iii) 17 selling price (iv) 18 selling price (units) (units) 21 000 units 0.2 4 200 19 000 units 0.2 3 800 19 000 units 0.5 9 500 17 500 units 0.5 8 750 16 500 units 0.3 4 950 15 500 units 0.3 4 650 18 650 17 200 96 DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY

Expected contribution 17 selling price ( 17 10.40) 18 650 123 090 18 selling price ( 18 10.40) 17 200 130 720 The existing selling price is 16, and if demand continues at 20 000 units per annum then the total contribution will be 112 000 [( 16 10.40) 20 000 units]. Using the expected value approach, a selling price of 18 is recommended. (b) Expected profit 130 720 85 100 fixed costs 45 620 Break-even point fixed costs ( 85 100)/contribution per unit ( 7.60) 11 197 units Margin of safety expected demand (17 200 units) 11 197 units 6003 units % margin of safety 6003/17 200 34.9% of sales Note that the most pessimistic estimate is above the break-even point. (c) An expected value approach has been used. The answer should draw attention to the limitations of basing the decision solely on expected values. In particular, it should be stressed that risk is ignored and the range of possible outcomes is not considered. The decision ought to be based on a comparison of the probability distributions for the proposed selling prices. For a more detailed answer see Probability distributions and expected value and Measuring the amount of uncertainty in Chapter 12. (d) Computer assistance would enable a more complex analysis to be undertaken. In particular, different scenarios could be considered, based on different combinations of assumptions regarding variable cost, fixed cost, selling prices and demand. Using computers would also enable Monte Carlo simulation (see Chapter 14) to be used for more complex decisions. (a) Possible Alternative outcomes Probability types of (level of of Payoff machine hire orders) outcomes ( 000) High High 0.25 2200 [(0.3 15 000) 2300] Medium 0.45 250 [(0.3 8500) 2300] Low 0.30 1100 [(0.3 4000) 2300] Medium High 0.25 1700 (0.3 15 000) 1500 1300 Medium 0.45 1050 [(0.3 8500) 1500] Low 0.30 300 [(0.3 4000) 1500] Low High 0.25 1350 (0.3 15 000) 1000 2150 Medium 0.45 700 (0.3 8500) 1000 850 Low 0.30 200 [(0.3 4000) 1000] Question 12.19 (b) Expected values: High hire level (0.25 2200) (0.45 250) (0.3 1100) 332 500 Medium hire level (0.25 1700) (0.45 1050) (0.3 300) 807 500 Low hire level (0.25 1350) (0.45 700) (0.3 200) 712 500 (c) Using the expected value decision rule, the medium hire contract should be entered into. Managers may be risk-averse, risk-neutral or risk-seeking. A risk-averse manager might adopt a maximin approach and focus on the worst possible outcome for DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY 97

each alternative and then select the alternative with the largest payoff. This approach would lead to the selection of the low initial hire level. A risk-seeking manager might adopt a maximax approach and focus on the best possible outcomes. This approach would lead to choosing the high initial hire contract, since this has the largest payoff when only the most optimistic outcomes are considered. (d) With perfect information, the company would select the advance plant and machinery hire alternative that would maximize the payoff. The probabilities of the consultants predicting high, medium and low demand are respectively 0.25, 0.45 and 0.30. The expected value calculation with the consultant s information would be: Advance Expected hire Payoff value level ( 000) Probability ( 000) High market high 2200 0.25 550 Medium market medium 1050 0.45 472.5 Low market low 200 0.30 60 1082.5 Expected value with consultant s information 1 082 500 Expected value without consultant s information 807 500 Maximum amount payable to consultant 275 000 Question 12.20 (a) Selling price 70 80 90 Maximum demand 75 000 60 000 40 000 Maximum revenue 5 250 000 4 800 000 3 600 000 Total variable cost 3 750 000 3 000 000 2 000 000 Fixed costs 800 000 800 000 800 000 R & D cost 250 000 250 000 250 000 4 800 000 4 050 000 3 050 000 Estimated profit 450 000 750 000 550 000 ( 000) 7000 5000 3000 Totalrevenue Total cost 1000 At 80 selling price Output 0 10 20 30 40 50 60 70 80 Optimal output level Figure Q12.20 98 DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY

The above analysis is based on the maximum sales demand. On this basis, the analysis indicates that profits are maximized at an output level of 60 000 units when the selling price is 80. It is preferable to use the most likely demand level and to incorporate uncertainty around the most likely demand into the analysis. (b) For a selling price of 90 there are three different demand levels, and for each demand level there are three different outcomes for actual unit variable cost. Therefore there are nine possible outcomes. The contribution and probability of each outcome is presented in the following schedule: (1) (2) (3) (4) (5) (6) (7) (8) Weighted Unit Total Joint outcome Demand variable Unit contribution probability (6 7) (000) Probability cost Probability contribution ( 000) (2 4) ( 000) 20 0.2 60 0.2 30 600 0.04 24.00 20 0.2 55 0.7 35 700 0.14 98.00 20 0.2 50 0.1 40 800 0.02 16.00 35 0.7 60 0.2 30 1050 0.14 147.00 35 0.7 55 0.7 35 1225 0.49 600.25 35 0.7 50 0.1 40 1400 0.07 98.00 40 0.1 60 0.2 30 1200 0.02 24.00 40 0.1 55 0.7 35 1400 0.07 98.00 40 0.1 50 0.1 40 1600 0.01 16.00 1.00 1121.25 Expected total contribution 1 121 250 Fixed costs 1 050 000 Expected profit 71 250 (c) To compare the three selling prices, it is necessary to summarize the information in part (b) for a 90 selling price in the same way as part (c) of the question. Note that fixed costs are deducted from the total contribution column in the schedule presented in (b) to produce the following statement: Prices under review 70 80 90 Probability of a loss Greater than or equal to 500 000 0.02 0 0 300 000 0.07 0.05 0.18 100 000 0.61 0.08 0.20 0 0.61 0.10 0.34 Probability of a profit Greater than or equal to 0 0.39 0.91 0.80 100 000 0.33 0.52 0.66 300 000 0.03 0.04 0.15 500 000 0 0.01 0.01 Expected profit Loss ( 55 750) 68 500 71 250 The following items should be included in the memorandum: (i) The 90 selling price has the largest expected profit, but there is also a 0.34 probability of not making a profit. (ii) Selling price of 80 may be preferable, because there is only a 0.10 probability of not making a profit. A selling price of 80 is least risky, and the expected value is only slightly lower than the 90 selling price. DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY 99

(iii) Subjective probability distributions provide details of the uncertainty surrounding the estimates and enable the decision-maker to select the course of action that is related to his personal risk/profit trade-off (see Chapter 12 for an explanation of this). (iv) Subjective probabilities are subject to all the disadvantages of any subjective estimate (e.g. bias). (v) Calculations are based on discrete probabilities. For example, this implies that there is a 0.7 probability that demand will be exactly 35 000. A more realistic interpretation is that 35 000 represents the mid-point of demand falling within a certain range. (d) If the increase in fixed costs represents an additional cost resulting from an increase in volume then this incremental cost is relevant to the pricing decision. If the fixed costs represent an apportionment then it is not relevant. Nevertheless, we noted in Chapter 11 that selling prices should be sufficient to cover the common and unavoidable long-run fixed costs. The research and development expenditure is a sunk cost, and is not a relevant cost as far as the pricing decision is concerned. However, the pricing policy of the company may be to recover the research and development expenditure in the selling price. The amount recovered per unit sold should be a policy decision. Note that the decision to write off research and development in one year instead of three will affect the reported profits. Question 12.21 (a) The calculations of the product variable costs per unit are: Newone Newtwo Labour and materials 82 44 Variable overheads 6 (6 hrs 1) 2 (2 hrs 1) Unit variable cost 88 46 Low-price alternative: The contributions per unit are 32 for Newone ( 120 88) and 14 ( 60 46) for Newtwo. The probability distributions are as follows: Newone Newtwo Demand Probability Contribution Demand Probability Contribution 1000 0.2 32 000 3000 0.2 42 000 2000 0.5 32 000 a 3000 0.5 42 000 3000 0.3 32 000 a 3000 0.3 42 000 Note a Machine capacity restricts outputs to 1000 units of Newone and 3000 units of Newtwo. Note that estimates indicate with 100% certainty that Newone will yield a contribution of 32 000 and Newtwo will yield a contribution of 42 000. Higher price alternative: The contributions per unit are 42 for Newone ( 130 88) and 24 ( 70 46) for Newtwo. The probability distributions are as follows: Newone Newtwo Demand Probability Contribution Demand Probability Contribution 500 0.2 21 000 1500 0.2 36 000 1000 0.5 42 000 2500 0.5 60 000 1500 0.3 42 000 a 3500 0.3 72 000 a Expected value 37 800 Expected value 58 800 100 DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY

Note a Output is restricted to 1000 units of Newone and 3000 units of Newtwo. Recommendations The above probability distributions indicate that Newtwo is preferable to Newone, irrespective of which price is set. At the higher selling price Newtwo yields a higher expected value. There is only a 0.2 probability that a lower contribution will be earned if the higher price is selected in preference to the lower price. The advantage of the lower price is that the outcome is certain, but, given the high probability (0.8) of earning higher profits with the higherprice alternative, a selling price of 70 is recommended. With the higher-price alternative, there is a 0.70 probability that machine hours will not be utilized. Any unused capacity should be used to sell Newone at 130 selling price. (b) Decision problems require estimates of changes in costs and revenues for choosing alternative courses of action. It is therefore necessary to distinguish between fixed and variable costs. Regression analysis can be used to estimate a cost equation, and tests of reliability can be applied to ascertain how reliable the cost equation is in predicting costs. For a description of regression analysis and tests of reliability you should refer to Chapter 24. A common test of reliability is the coefficient of determination, which can be calculated by squaring the correlation coefficient. The coefficient of determination for the cost equation used in the question is 0.64 (0.8 2 ). Consequently, 36% of the variation in cost is not explained by the cost equation used in the question. It is possible that activity bases other than machine hours might provide a better explanation of the relationship between costs and activities. Alternatively, changes in costs might be a function of more than one variable. In such circumstances, cost equations based on multiple regression techniques should provide more reliable cost estimates. The following is a decision tree relating to the question: Question 12.22 (1) MD s price (1) CN s price 1.2 (1) KL s price 1.2 1.1 P = 0.6 P = 0.2 P = 0.1 Decision tree 1.1 P = 0.4 P = 0.3 (4) (5) Expected Unit sales (millions) contribution (6) Total contribution ( m) (7) Joint probability (2 3) (8) Expected value (6 7) ( m) 2.7 0.80 2.16 0.02 0.0432 2.3 0.80 1.84 0.12 0.2208 2.2 0.80 1.76 0.06 0.1056 1.1 P = 0.3 2.4 0.80 1.92 0.12 0.2304 1.2 P = 0.4 P = 0.7 2.2 0.80 1.76 0.28 0.4928 P = 1.0 2.1 0.80 1.68 0.4 0.6720 1.1 1.1 1.1 P = 0.3 P = 0.3 1.0 1.7648 2.8 0.70 1.96 0.09 0.1764 P = 0.7 P = 0.7 2.4 0.70 1.68 0.21 0.3528 P = 1.0 P = 1.0 P = 1.0 2.3 0.70 1.61 0.70 1.1270 1.0 1.6562 2.9 0.60 1.74 1.0 1.74 Figure Q12.22 DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY 101

The variable cost per litre is as follows: Direct materials 0.12 Direct wages 0.24 Indirect wages etc. (16 % 0.24) 0.04 0.40 and the range of contributions are: 0.80 for a selling price of 1.20 0.70 for a selling price of 1.10 0.60 for a selling price of 0 The decision tree indicating the possible outcomes presented in Figure Q12.22 shows that the expected value of the contribution is maximized at a selling price of 1.20. Fixed costs are common and unavoidable to all alternatives, and are therefore not included in the analysis. However, management might prefer the certain contribution of 1.74 million at a selling price of 0. From columns 6 and 7 of the decision tree it can be seen that there is a 0.60 probability that contribution will be in excess of 1.74 million when a selling price of 1.20 is implemented. The final decision depends on management s attitude towards risk. Question 12.23 (a) Budgeted net Profit/Loss outcomes for year ending 30 June Client Fee per Variable cost Contribution Total contrib. Days Client day per client day per client day per year 15 750 180 95 85 1 338 750 15 750 180 85 95 1 496 250 15 750 180 70 110 1 732 500 13 125 200 95 105 1 378 125 ` 13 125 200 85 115 1 509 375 13 125 200 70 130 1 706 250 10 500 220 95 125 1 312 500 10 500 220 85 135 1 417 500 10 500 220 70 150 1 575 000 (b) The maximax rule looks for the largest contribution from all outcomes. In this case the decision maker will choose a client fee of 180 per day where there is a possibility of a contribution of 1 732 500. The maximin rule looks for the strategy which will maximize the minimum possible contribution. In this case the decision maker will choose client fee of 200 per day where the lowest contribution is 1 378 125. This is better than the worst possible outcome from client fees per day of 180 or 220 which will provide contribution of 1 338 750 and 1 312 500 respectively. The minimax regret rule requires the choice of the strategy which will minimize the maximum regret from making the wrong decision. Regret represents the opportunity lost from making the wrong decision. The calculations in part (a) are used to list the opportunity losses in the following regret matrix: State of nature Low variable Most likely variable High variable cost of 70 cost of 85 cost of 95 Choose a fee of 180 0 13 125 39 375 Choose a fee of 200 26 250 0 0 Choose a fee of 220 157 500 91 875 65 625 102 DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY

At a variable cost of 70 the maximum contribution is 1 732 500 derived from a fee of 180. Therefore there will be no opportunity loss. At a fee of 200 the opportunity loss is 26 250 ( 1 732 500 1 706 250) and at the 220 fee the opportunity loss is 157 500 ( 1 732 500 1 575 000). The same approach is used to calculate the opportunity losses at variable costs of 85 and 95. The maximum regrets for each fee are as follows: 180 39 375 200 26 250 220 157 500 The minimum regret is 26 250 and adopting a minimum regret strategy will result in choosing the 200 fee per day alternative. (c) The expected value of variable cost = 95 0.1 + 85 0.6 + 70 0.3 = 81.50 For each client fee strategy the expected value of budget contribution for the year is calculated as follows: * fee of 180 : 15 750 (180 81.50) = 1 551 375 * fee of 200 : 13 125 (200 81.50) = 1 555 312.50 * fee of 220 : 10 500 (220 81.50) = 1 454 250 A client fee of 200 per day is required to give the maximum expected value contribution of 1 555 312.50. Note that there is virtually no difference between this and the contribution where a fee of 180 per day is used. (d) Profit can be increased by making cost savings provided that such actions do not result in a fall in demand and a reduction in revenues. Alternatively, investments may be made that will increase the level of service and thus demand. Profits will increase if the extra revenues exceed the increase in costs. The balanced scorecard approach to performance measurement and the determinants of performance measurement relating to service organizations described in Chapter 23 can be used to identify appropriate performance areas for the health centre. The performance areas identified in Exhibit 23.5 in Chapter 23 include quality of service, flexibility, resource utilization and innovation. Each of these areas is discussed below. (i) Quality of service may be improved by upgrading facilities such as a cafeteria, free daily newspapers and better waiting room facilities. This may increase demand and generate additional revenues which exceed the cost increases. (ii) Flexibility of service may be improved by providing additional sports/exercise facilities that are not currently available. In addition, additional exercise and dietary consultants who can provide services that are not currently available. (iii) Resource utilization may be improved by better scheduling relating to the use of the exercise equipment and staff time and extending the opening hours. The aim should be to provide at least the same level of service with fewer resources. (iv) Innovation may take the form of new services such as an extension of the range of health advice that can be provided and introducing on-line booking systems which can be directly accessed by the clients. DECISION-MAKING UNDER CONDITIONS OF UNCERTAINTY 103